Methods and systems to quantify and index correlation risk in financial markets and risk management contracts thereon

ABSTRACT

Systems and methods for creating indicators to quantify and index correlation risk that is market-wide among a broad set of asset classes or portfolio specific relative to an investor&#39;s portfolio holdings. The present disclosure relates to risk management in financial markets, and in particular to systems and methods for quantifying and indexing correlation risk such that these indices can serve as underlying assets for futures and options or other financial instruments that investors would use to hedge against the risk.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 16/905,542, filed Jun. 18, 2020, which itself is a continuation of U.S. patent application Ser. No. 13/677,278, filed Nov. 14, 2012, which claims the benefit of U.S. Provisional Patent Application No. 61/629,227, filed Nov. 14, 2011, the contents of which are incorporated by reference herein in their entireties.

This application is also related to U.S. patent application Ser. No. 13/677,273, entitled “METHODS AND SYSTEMS TO CREATE SYNTHETIC VARIABLES, INDICATORS AND INDICES FROM MULTIPLE DATA AND METADATA SOURCES AND RISK MANAGEMENT CONTRACTS THEREON,” U.S. patent application Ser. No. 13/677,276, entitled “METHODS AND SYSTEMS TO EXTRACT SIGNALS FROM LARGE AND IMPERFECT DATA SETS,” U.S. patent application Ser. No. 13/677,277, entitled “METHODS AND SYSTEMS TO QUANTIFY AND INDEX LIQUIDITY RISK IN FINANCIAL MARKETS AND RISK MANAGEMENT CONTRACTS THEREON,” and U.S. patent application Ser. No. 13/677,297, entitled “METHODS AND SYSTEMS TO QUANTIFY AND INDEX SENTIMENT RISK IN FINANCIAL MARKETS AND RISK MANAGEMENT CONTRACTS THEREON,” all of which were filed on Nov. 14, 2012 and are incorporated by reference herein in their entireties.

TECHNICAL FIELD

The present disclosure relates to risk management in financial markets, and in particular to systems and methods for quantifying and indexing risks such that these indices can serve as underlying assets for futures and options and other financial instruments that investors would use to hedge against the risks.

BACKGROUND

Markets are incomplete, in that, it is not possible to hedge against all potential risks. Recent financial crises have highlighted the need for more effective risk management. Portfolio managers are increasingly faced with the need to better understand and efficiently manage multiple sources of risk that can affect the value of their holdings. This can be particularly challenging for holders of multiple asset classes across multiple geographies. Some customized hedging solutions are available to professional money managers, such as, for example the use of swaps. But these over-the-counter instruments are unregulated, expensive, illiquid, and carry significant counter-party risk. The over-the-counter swaps market operates in the shadows of the financial markets, with an estimated size of $650 billion. (See, e.g., http//www.huffingtonpost.com/2012/07/08/us-derivatives-reform-rules_n_1656980.html, which is hereby incorporated by reference herein in its entirety.) Indeed, the lack of transparency in the swaps market is largely blamed in the collapse of financial firms such as Lehman Brothers and insurer American International Group during the financial crisis of 2007-2009, which led to billions of dollars in government bailouts, a burden ultimately shouldered by taxpayers.

The specter of regulation looms over the derivatives market. The 2010 Dodd Frank financial reform law is meant to increase transparency in order to mitigate systemic risk, but compliance with such regulation will be expensive, and many small traders will be likely shut out of the market. Additionally, customized and complex hedging solutions through the use of swaps and other derivatives have long been out of reach for individual investors, and costly regulation will further prohibit individual investors from being able to hedge their portfolios from serious risks that can devastate the value of their portfolios. Recent decades have brought technological advances that democratized equity trading for individual investors by making online trading accessible and affordable, but effective risk management remains out of reach.

Risk management must be simplified and democratized in order to build and preserve wealth, both for institutions as well as for individuals. Risk metrics and risk management contracts must be accessible, affordable, and transparent. Improved risk management techniques will assist in mitigating the boom-bubble-bust cycles that have roiled financial markets in recent decades.

One example of improvement in risk management techniques was the introduction of the Chicago Board Options Exchange Market Volatility Index, also known by its ticker symbol, “VIX”. The VIX is a popular measure of the implied volatility of S&P 500 index options. It is often referred to as the fear index or the fear gauge, because it represents one measure of the market's expectation of stock market volatility over the subsequent 30-day period. The concept of a volatility index, and financial instruments based on such an index, was first proposed by Menachem Brenner and Dan Galai in 1986, and was published in “New Financial Instruments for Hedging Changes in Volatility,” appearing in the July/August 1989 issue of Financial Analysts Journal. (See, e.g., http//people.stern.nyu.edu/mbrenner/research/FAJ_articleon_Volatility_Der.pdf, which is hereby incorporated by reference herein in its entirety.)

While stock index options and futures give investors the ability to hedge against market and interest rate volatility, the VIX allows investors to hedge against the risk of changes in volatility. Changes in market volatility can be brought about by macroeconomic factors such as inflation or economic policy, or by firm-specific factors such as changes in capital structure or news about performance. The ability to hedge against changes in volatility has helped to complete the market by providing insurance against a very real and potentially devastating portfolio risk.

But markets remain significantly incomplete. Investors today are faced with a multitude of serious risks that remain uninsurable. These risks are frequently discussed by market practitioners and in the financial media, but they are discussed as broad concepts, often in nebulous terms. As of yet, there has not been a concerted effort to quantify and index many of these risks so that efficient and accessible hedging methods can be introduced.

There are three risks in particular that are of vital importance to investors participating in modem financial markets: 1) correlation risk; 2) liquidity risk; and 3) sentiment risk. We propose systems and methods to quantify and index these risks, and risk management contracts in order to insure against these risks. These indices would serve as underlying assets for futures and options and other financial instruments that investors would use to hedge against the risk of changes in correlation, liquidity, and sentiment in financial markets.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example network environment in which various exemplary embodiments of the present disclosure can operate;

FIG. 2 is a schematic diagram illustrating the method for creating a correlation risk index; and

FIG. 3 depicts an exemplary graphical display of a correlation risk index over time.

SUMMARY

Systems and methods for creating indicators to quantify and index correlation risk that is market-wide among a broad set of asset classes, a collection of assets or securities, securities in an index, or portfolio specific relative to an investor's portfolio holdings. The present disclosure relates to risk management in financial markets, and in particular to systems and methods for quantifying and indexing correlation risk such that these indices can serve as underlying assets for futures and options and other financial instruments that investors would use to hedge against the risks.

In accordance with some embodiments, a method for providing a risk index is provided, the method comprising: selecting a plurality of assets; retrieving price data associated with each of the plurality of assets over a time window; calculating returns over the time window; determining a plurality of correlation measures and correlation and risk indicators between the plurality of assets over the time window using the retrieved price and/or calculated returns data, wherein a plurality of weights are assigned to the plurality of assets; generating a correlation risk index for the plurality of assets based on the plurality of correlation measures and correlation risk indicators; and providing the correlation risk index.

In accordance with some embodiments, a system for providing a risk index is provided, the system comprising a hardware processor that is configured to: select a plurality of assets; retrieve price data associated with each of the plurality of assets over a time window; calculating returns over the time window; determine a plurality of correlation measures and correlation risk indicators between the plurality of assets over the time window using the retrieved price and/or calculated returns data, wherein a plurality of weights are assigned to the plurality of assets; generate a correlation risk index for the plurality of assets based on the plurality of correlation measures and correlation risk indicators; and provide the correlation risk index.

In accordance with some embodiments, a non-transitory computer-readable medium containing computer-executable instructions that, when executed by a processor, cause the processor to perform a method for providing a risk index, is provided. The method comprising: selecting a plurality of assets; retrieving price data associated with each of the plurality of assets over a time window; calculating returns over the time window; determining a plurality of correlation measures and correlation risk indicators between the plurality of assets over the time window using the retrieved price data and/or calculated returns data, wherein a plurality of weights are assigned to the plurality of assets; generating a correlation risk index for the plurality of assets based on the plurality of correlations measures and correlation risk indicators; and providing the correlation risk index.

In some embodiments, the method can include: alerts, displays, graphs, GUIs, comparisons, trends over time, whether the correlation risk index is increasing or decreasing, a scrolling ticker of correlation risk index levels and changes (numerical and percent change), price impact of correlation risk on securities, predictive power of correlation risk for returns, etc.

In some embodiments, the alerts can be visual (e.g., a light bulb changing colors), graphical on screen, text coming as a text message or email alert, a scrolling ticker alert, or auditory (e.g., chimes, bells).

In some embodiments, the alerts can be based on some user-configured threshold level or rate of increase or decrease, abnormally high or low levels, sudden changes, etc.

In some embodiments, the correlation of security returns can be percent change or log returns.

In some embodiments, the correlation can be Spearman, Pearson, Kendall, time-varying, dynamic, or stochastic or any other way to measure correlation or cointegration or co-movement or dependence between assets.

In some embodiments, the method can determine changes in correlation risk over time.

In some embodiments, the method can determine market-wide or portfolio specific correlation risk, correlation risk of asset classes or a collection of assets (stocks, bonds, commodities, currencies, real estate, art, wine, stamps, any asset type) or a collection of securities or assets, securities in a well known market index, securities in a portfolio, investment style metrics (size, book-to-market, momentum, etc.), or company fundamental measures like price-to-earnings ratios.

In some embodiments, the method further comprises a scenario analysis of what would happen to level of correlation risk in a portfolio if certain trades are made or assets substituted, or if holdings of certain assets increase or decrease; and in addition a securities screening and selection module to identify candidate securities to buy or sell on the basis of correlation risk.

In some embodiments, the method further comprises analyzing portfolio holdings from an online brokerage account or manually entered by a user, or uploaded via file transfer in a format such as Excel or ascii.

In some embodiments, the method further comprises displaying correlation risk index that (i) trends over time, (ii) varies by collection of securities or assets or portfolio holdings, or (iii) concurrently trends over time and varies by collection of securities or assets or portfolio holdings.

In some embodiments, weights can be determined by any signal extraction method or algorithm, such as Wiener-Kolmogorov, Kalman Filter, Modified Principal Components Analysis-Kalman Filter, Principal Components Analysis (static, rolling, or recursive). Rolling of recursive Principal Components Analysis done on a standard correlation or covariance matrix, or a matrix of pair-wise correlations or covariances, performed sequentially after an initial ramp-up window, such that a time series of eigenvector loadings is generated, of the first and/or the n-th principal component, forming a time series of weights that are then applied to the correlations of the returns or price changes between the n assets.

In some embodiments, the method further comprises:

-   -   1) retrieving price data;     -   2) calculating returns on n assets;     -   3) determining n(n−1)/2 unique asset pairs;     -   4) calculating correlation (or covariance or any other measure         of dependence) of each unique asset pair over some initial time         period (e.g., 6 months), specifying an initial window of k         observations;     -   5) expanding the window by one observation each time period         (k=k+1) and calculating a running measure of correlation (the         window grows one observation at a time but is anchored at the         first observation; alternatively could be a rolling correlation         calculation, moving average, etc.) and obtain a time series of         correlations;     -   6) determining initial ramp-up period (e.g., 20 months) and         computing a standard correlation or covariance matrix, or         alternatively, a matrix of pairwise correlations or covariances;     -   7) performing the first eigenvalue decomposition on the matrix         obtained in Step 6, and begin recursive calculation;     -   8) at each time period, computing a new correlation matrix, over         n=n+1 observations;     -   9) performing new eigenvalue decomposition at each time period,         and collecting the factor loadings from the first or the n-th         principal component at each time period     -   10) generating a time series of eigenvector loadings, these         become the weights for the index at each point in time.     -   11) perform the calculation at each time period as follows:

${{\sum\limits_{1 \leq i < j \leq n}^{{n({n - 1})}/2}{w_{i,j}r_{i,j}*M}} + S} = {{CRI}\left( {{Correlation}{Risk}{Index}} \right)}$ where n is the number of assets, n(n−1)/2 is the number of unique asset pairs from n assets where 1≤i<j≤n, w_(i,j) is the weight corresponding to the correlation measure for the i-th and j-th assets in a unique asset pair, r_(i,j) is the correlation measure between the i-th and j-th assets in a unique asset pair, M is a multiplier and S is a scalar. It is understood that the multiplier M and scalar S may take on any value, including 1 and 0, respectively. Alternatively, the weights could be arbitrarily determined, or they could be weights such as market-capitalization weights like the S&P index weights, or portfolio weights for a portfolio-specific correlation risk index.

In some embodiments, the method determines the correlation risk index trends over time, against other financial market benchmark indexes, economic indicators, or other risk indexes.

In some embodiments, the method determines the predictive power of the CRI by doing econometric tests, such as a cross-sectional multi-factor model, or granger causality tests against returns.

In some embodiments, a method of tracking securities to track the value of the Correlation Risk Index is provided, where financial instruments, such as futures, options, and options on futures can be issued to speculate on increasing or decreasing values of the indicators. Indices can be created from the indicators to quantify certain portfolio risks or risks in the economy—for example, correlation risk—and the derivatives or other financial instruments can be used to provide a method to hedge against those risks.

In some embodiments, the method further comprises calculating the moments of data series, wherein such calculations determine the statistical properties of the data, comprising one or more of the: (i) mean, (ii) median, (iii) mode, (iv) variance, (v) standard deviation, (vi) kurtosis, or (vii) skewness, and using these as elements of the metric.

In some embodiments, the method further comprises calculating the statistical relationships between data items, comprising one or more of (i) correlation, (ii) cointegration, or (iii) covariance, and using these as elements of the metric.

In some embodiments, a method of creating an indicator of correlation risk, comprising: selecting N assets or asset classes of interest or N securities; determining a time period over which correlation is relevant; obtaining price data for the N assets or asset classes or N securities over the time period of interest; calculating returns over the time period of interest; calculating correlation measures between the N assets or asset classes or N securities over the relevant time period; calculating correlation between the returns or price changes of the assets or asset classes in real-time, at each point in time; determining weights of each of the correlations either arbitrarily or via a signal extraction algorithm such as a rolling or recursive Principal Components Analysis or the novel MPCA-KF algorithm or any other type of signal extraction algorithm; and forming a composite correlation risk index.

In some embodiments, a method of hedging correlation risk, the method comprising: issuing derivatives—options, futures, options on futures—or ETFs to track the value of the composite correlation index, and issuing derivatives—options, futures, options on futures—or ETFs to track the value of each of the underlying correlation of asset pairs among the n assets.

In some embodiments, as the price of each underlying derivative contract on correlation of asset pairs changes, the price of the composite index changes in real time as the prices of the underlying assets change.

In some embodiments, derivative instruments on style factors and other investment discipline metrics are provided, wherein said instruments include at least one of futures, options, options on futures, and ETFs.

In some embodiments, a method of creating synthetic style factor tracking indexes to mimic style portfolios is provided, comprising: selecting a style factor to be mimicked (such as momentum, size (Small capitalization minus Big capitalization), growth or value (High book-to-market value minus Low book-to-market value), mean reversion, or contrarian); creating a hypothetical portfolio according to a set of rules that conform to the investment discipline or style; conducting hypothetical trades on the portfolio according to the set of rules belonging to the style; and calculating returns on the hypothetical trades belonging to the hypothetical portfolio; creating an index to track the hypothetical returns on the hypothetical portfolio which has been hypothetically managed according to the rules belonging to the investment style.

In some embodiments, a method of hedging synthetic style factors, comprising: issuing derivatives to track the value of the style factor index, wherein said derivatives include at least one of options, futures, options on futures and ETFs.

In some embodiments, an index showing the correlation risk among style factors and risk management contracts thereon is provided.

DETAILED DESCRIPTION A. Overview

In exemplary embodiments of the present disclosure, systems and methods for creating a correlation risk index are presented.

Returns on asset classes such as equities, commodities, and bonds tend to be correlated. This significantly impedes diversification efforts, since correlated assets move in tandem. The result is that investors have “no place to hide” when correlations are high. Exacerbating this is the fact that correlations tend to rise in falling markets, meaning that even the most highly diversified portfolios will suffer significant losses when markets face increased volatility.

One solution to this problem is to create an index that measures and tracks asset class (securities) price or return correlation. Such an index can be generalized to track the co-movement of broadly defined asset classes, such as the correlations of returns between equities, gold, oil and bonds, for example. Similarly, such an index can be highly specialized to track individual portfolio holdings, such as the co-movement of individual stocks, ETFs, particular issues of bonds, options with particular strike prices or expiration dates, and so on.

The indicators can be constructed using price data on securities, from which returns and correlations over a certain period of time are calculated. Similarly, such an index can be constructed using other data, such as volume, or company fundamental data. An index can be constructed using either (i) a simple method, such as, for example, a weighted average whereby the weights can be arbitrarily assigned, or weighted by market capitalization which mimics index weights such as the market-capitalization weights that are applied to the S&P 500, or by applying a divisor such as that used to calculate the Dow-Jones Industrial Average, or by (ii) mathematical formulae, transformations, statistical formulae, or algorithmic methods, such as optimization or signal extraction methods such as Principal Components Analysis or another signal extraction method.

The term “data” as used herein also includes (i) the moments of the statistical distribution of the data, such as, for example, the mean, standard deviation, variance, standard deviation, kurtosis, and skewness; (ii) statistical measures of the data, such as the median and mode; (iii) transformations such as, for example, arithmetic or logarithmic; or (iv) properties of the data over time, such as, for example, a moving average. It is also understood that the term “data” further includes various statistical relationships amongst the various sources of data, such as, for example, correlations and covariances. The term data as used herein also includes technical analysis data such as relative strength indicators (RSI), moneyflow, and other price and volume data.

The term “data” is also understood herein to include so-called “style factors”, which can measure the return achieved from holding a portfolio of stocks that have been selected according to certain criteria, discipline or investment style. Such factors relate to a company's size (i.e., market capitalization) and book value (i.e., the net asset value of a company, calculated by total assets minus intangible assets (patents, goodwill) and liabilities). That is, such factors relate to whether a stock is a small-capitalization or large capitalization stock, and whether the stock is considered to be cheap, i.e., a value stock, or expensive, i.e., a growth stock. Other style factors include momentum, growth at a reasonable price (GARP), mean reversion, or contrarian. Style factors can include, among others, the RMRF, SMB, and HML factors popularized by Fama and French (Fama, Eugene F.; French, Kenneth R. (1993), in “Common Risk Factors in the Returns on Stocks and Bonds”. Journal of Financial Economics 33 (1): 3-56 and Fama, Eugene F.; French, Kenneth R. (1992). “The Cross-Section of Expected Stock Returns”. Journal of Finance 47 (2): 427-465.) and the momentum factor (MOM) popularized by Carhart (Carhart, Mark M. (1997). “On Persistence in Mutual Fund Performance”, The Journal of Finance, 52 (1): 57-82.). A style factor measures or tracks the return on a portfolio where the assets have been selected according to a certain investment approach, style or theory.

In exemplary embodiments of the present disclosure, a synthetic index can be created to track the performance of a style factor, and correlations or covariances between style factors. Further, derivatives—options, futures, and options on futures—or Exchange Traded Funds, can be issued on the value of the index, giving market participants a way to hedge the risk associated with the performance of a certain investment style, including the correlation of various investment styles. Derivative instruments can be issued on style factors and other investment discipline metrics, wherein said instruments include at least one of futures, options, options on futures, and ETFs.

It is noted that financial instruments on an index which tracks a certain investing style is a more efficient and economical way to gain portfolio exposure to an investment style, rather than investing directly in a portfolio of individual assets that meet that investment style or criteria. If an individual investor were to try to manage a portfolio of a given style, it would be very expensive to replicate that style because it would require an extremely large number of stocks, which would be onerous in terms of (1) the capital required to hold multiple positions, (2) the transaction costs, (3) the time required to manage the portfolio, and (4) the knowledge required to manage the portfolio. Thus, most individual investors are not able to hold portfolios that accurately adhere to a certain investment discipline. Alternatively, an individual investor could buy shares of a mutual fund that is managed according to a certain investing style or approach, but this can also be expensive because of the fees charged by mutual funds. It is noted that mutual funds are not and will not be supportive of (i) an index that tracks a certain investing style, and (ii) risk management contracts traded on that index, because investors would have no incentive to buy shares of the mutual fund if they could just buy shares in a synthetic portfolio represented by a style index, or derivative contracts on a style index. A style index would be a much more economical way for an investor to gain exposure to a certain investment style without incurring the costs to recreate the style portfolio or pay mutual fund fees. It is noted that the technology required to calculate returns on a style factor and create a tracking index in real time has not been widely available until recently. Therefore, indices on style factors and derivative contracts on those indices have not been commercialized due to the significant political and technological barriers.

In exemplary embodiments of the present disclosure, once generated as described herein, such indicators can be used to identify and quantify correlations that affect the economy and/or asset prices and/or securities comprising a benchmark index. In this case, the indicator becomes an index, whose value changes in real time. Thus, financial instruments—i.e., risk management contracts in the form of futures, options, and options on futures or Exchange Traded Funds (ETFs) or other financial instruments—can be introduced which track the value of such an index. This can provide financial market participants with a method of hedging certain risks that are currently neither quantified nor hedgeable.

Data, such as asset or securities prices and other data can be obtained from stock exchanges, or any number of sources including:

-   -   IHS Global Insight, Bloomberg, Reuters, Capital IQ, CME Group         COMEX, S&P Capital IQ, Chicago Board of Trade, Chicago Board of         Options Exchange, TAQ Trade and Quote Data, NYMEX, Standard &         Poors, NYSE Euronext, NASDAQ, I/B/E/S (Institutional Brokers'         Estimate System), Zacks Investment Research, or similar sources.

In exemplary embodiments of the present disclosure an exemplary indicator can be tested against various types of historical data by means of (i) a simple statistical analysis, such as correlation or covariance, or by (ii) an econometric model such as Ordinary Least Squares, specified by (y=a+bx+e) whereby the indicator would obtain associated parameter estimates, such as Alpha (a, a constant), Beta (b, the covariance between x and y divided by the variance of x), the Error (e), sigma (the standard deviation of x) and sigma-squared (the variance of x). Other methods to estimate an econometric model can include, for example, General Method of Moments, Maximum Likelihood Estimate, etc. An indicator can be modeled either linearly or non-linearly. It is recognized that in several instances the data may be of different sampling frequencies, so that either the data must be converted to the same frequency, or a technique for modeling mixed frequency data must be employed such as, for example, a MIDAS (Mixed Data Sampling) regression.

In exemplary embodiments of the present disclosure indicators can be made available on a platform which allows users to (i) specify data inputs for creating custom correlation risk indicators, (ii) apply a mathematical formula, statistical method, or signal extraction algorithm to calculate the indicator, (iii) test the indicator against actual outcomes and historical data such as portfolio data, (iv) make forecasts of future values of the correlation of assets to economic data or asset prices, (v) conduct scenario analysis for how the level of correlation risk would change in a portfolio with the addition or deletion of holdings, (vi) screen and select candidate securities to buy or sell on the basis of correlation risk. The indicators can be displayed numerically, or in graphical form, and can, for example, be compared to one another, displayed in real-time or as time series historical data, and/or compared to historical outcome data to display trends and co-movements over time. In exemplary embodiments of the present disclosure such indicators can be used to forecast future outcomes and predict future values of various economic or financial conditions, quantities, or states, or asset prices.

Thus, various exemplary embodiments of the present disclosure can include one or more of the following processes, systems or methods:

1) The use of asset prices and/or returns and/or other data to make indicators regarding current and/or future correlation risks of economic or financial conditions, quantities or states, or asset prices;

2) Combining several of these data sources with (i) other asset price and/or returns data, and/or (ii) economic, financial and news data, or combining several sources of economic, financial and news data to create a composite indicator of current and/or future economic or financial conditions, quantities or states, or asset prices; in such combinations the combination weights can be determined either arbitrarily or by applying a signal extraction algorithm. 3) An analysis platform for statistical and econometric models combining correlation data with economic and financial historical and real-time data sources to generate parameter estimates and make forecasts of future values of portfolios of assets, correlations, economic or financial data or predict asset returns; and 4) Creation of an index based on the correlation risk indicators and tradable financial instruments based on the value of the index, such that if an indicator quantifies a certain risk factor, the derivative instruments (futures, options, options on futures) or Exchange Traded Fund (ETF) or other financial instruments provide a method of hedging the risk quantified by the indicator.

In exemplary embodiments of the present disclosure, where multiple data sources are chosen to construct an indicator, the data sources can be combined into a composite indicator by assigning weights to each data source after the data have been appropriately processed and transformed. It is understood that such weights range in value between −1 and 1, and collectively sum to a fixed number.

A brief review of correlation and correlation risk are presented, as well as a description of signal extraction methods, including principal components analysis.

Correlation and Correlation Risk

Correlation is a well-known concept for measuring the linear relationship between two or more variables. It plays a major role in a number of classical approaches in finance: the capital asset pricing model as well as arbitrage pricing theory (APT) rely on correlation as a measure for the dependence of financial assets. In the multivariate Black-Scholes model correlation of the log-returns is used as a measure of the dependence between assets. The main reason for the importance of correlation in these frameworks is that the considered random variables obey—under an appropriate transformation—a multivariate normal distribution. Correlation is moreover a key driver in portfolio credit models, and the term default correlation has been coined for this. Correlation as a measure of dependence fully determines the dependence structure for normal distributions and, more generally, elliptical distributions, while it fails to do so outside this class. Hence, approaches relying on multivariate Brownian motions and transformations thereof naturally determine the dependence structure via correlation. Extending this, there are a number of approaches generalizing the simple linear correlation to a time-varying (dynamic) and stochastic correlation, and related measures, such as the Pearson Product-Moment Correlation, Spearman's Rank-Order Correlation and Kendall's Tau Correlation.

Correlation risk refers to the risk of a financial loss when correlation in the market changes. It plays a central role in risk management and the pricing of financial instruments, such as basket derivatives. In risk management, correlation risk refers to the risk of a loss in a financial position occurring due to a difference between anticipated correlation and realized correlation. In particular, this occurs when the estimate of correlation was wrong or the correlation in the market changed. Asset correlation shows two typical stylized features. (See, e.g., F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81:637-654, 1973, which is hereby incorporated by reference herein in its entirety.) The first is correlation clustering, whereby periods of high (low) correlation are likely to be followed by periods of high (low) correlation. The second is asymmetry and co-movement with volatility. High volatility in falling markets goes hand in hand with a strong increase in correlation, but this is not the case for rising markets. (See, e.g., T. G. Andersen, T. Bollerslev, F. X. Diebold, and H. Ebense. The distribution of realized stock return volatility. Journal of Financial Economics, 61:43-76, 2001, which is hereby incorporated by reference herein in its entirety.) Analysis of the 1987 crash revealed that correlation risk is identified as a reason the co-movement of stock market declines and increasing volatility. (See, e.g., R. Rubinstein. Comments on the 1987 stock market crash: 11 years later. Investment Accumulation Products of Financial Institutions published by The Society of Actuaries, 2001, which is hereby incorporated by reference herein in its entirety.) Notably this reduces opportunities for diversification in stock-market declines. Correlation among assets in a portfolio is commonly referred to as the “nowhere to hide” problem, because assets that are correlated move in tandem.

Correlations play a central role in financial markets. There is considerable evidence that correlations between asset returns change over time and that stock return correlations increase when returns are low. A market-wide increase in correlations negatively affects investor welfare by lowering diversification benefits and by increasing market volatility, so that states of nature with unusually high correlations may be expensive.

For two random variables X and Y with finite and positive variances their correlation is defined as:

$\begin{matrix} {{{Corr}\left( {X,Y} \right)} = \frac{{Cov}\left( {X,Y} \right)}{{{Var}(X)}*{{Var}(Y)}}} \\ {{where}\text{}} \\ {{{Cov}\left( {X,Y} \right)} = {E\left\lbrack {\left( {X - {E(X)}} \right)\left( {Y - {E(Y)}} \right)} \right\rbrack}} \end{matrix}$ is the covariance of X and Y, and E is the expectations operator.

Some properties of correlation are the following:

Corr(X,Y) is a number in [−1,1], that is, correlation is bounded by −1 and 1, and it is equal to −1 or 1 if and only if X and Y are linearly related, i.e., Y=a+bX.

for constants a,b with b≠0.

Corr=−1 if b<0

Corr=1 if b>0

For constants a,b Corr(X+a,Y+b)=Corr(X,Y) If X and Y are independent, then Corr(X,Y)=0 If Corr(X,Y)=0, then X and Y are uncorrelated. If X is m-dimensional and Y is n-dimensional then Cov(X,Y) is given by the m×n matrix with entries Cov(X_(i),Y_(j)). Σ=Cov(X,X) is the covariance matrix. Σ is symmetric and positive semi-definite, i.e., x^(T)Σx≥0 for all x∈

^(m). Cov(a+BX,c+DY)=BCov(X,Y)D^(T) for a∈

^(o), c∈

^(p), B∈

^(o×m), D∈

^(p×n). Corr(X,X) has the entries Corr(X_(i),X_(j)), 1≤i≤m, 1≤j≤n. Correlation is invariant under linear increasing transformations such that Corr(a+bX,c+dY)=Corr(X,Y) if bc>0. If bc<0 only the sign of the correlation changes.

The correlation matrix of X is Corr(X,X). It is symmetric and positive semi-definite.

The risk management of a portfolio as well as portfolio optimization depends heavily on the correlation. For example, assume that a financial position is given by portfolio weights w₁, . . . , w_(d) and the distribution of the assets X is multivariate normal. Then the profit and loss (P&L) of the position is given by

${\sum\limits_{i = 1}^{d}{w_{i}X_{i}}},$ normally distributed with mean

$\sum\limits_{i = 1}^{d}{w_{i}{E\left( X_{i} \right)}}$ and variance

$\sum\limits_{i,{j = 1}}^{d}{w_{i}w_{j}{{Cov}\left( {X_{i},X_{j}} \right)}}$ which equals

$\sum\limits_{i,{j = 1}}^{d}{w_{i}^{2}{{Var}\left( X_{i} \right)}}$ if the positions are uncorrelated. Otherwise, the value-at-risk depends on the correlations of all assets and therefore a change in correlation may significantly alter the risk of the position or portfolio of assets. Hence, it is of vital importance to quantify and track the correlation risk affecting any portfolio of assets. Principal Components Analysis and Other Signal Extraction Methods

Principal components analysis is a technique used to reduce multidimensional data sets to lower dimensions for analysis. PCA allows for a large set of correlated variables to be employed together, without the problem of multicollinearity, as the extracted factors are pairwise orthogonal. Sensitivity to units of measurement is avoided by standardizing the variables to have mean zero and unit variance, before calculating the principal components. Principal components of the indicators are formed by extracting the characteristic root of the standardized correlation or covariance matrix of the variables. That is, an eigenvalue decomposition, or an eigendecomposition, is performed. In the mathematical discipline of linear algebra, eigendecomposition, or sometimes spectral decomposition, is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, either remain proportional to the original vector (i.e., change only in magnitude, not in direction) or become zero. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector changes when multiplied by the matrix. The eigenvectors are sometimes also called characteristic vectors or latent vectors. Similarly, the eigenvalues are also known as characteristic values, characteristic roots or latent roots. The resulting eigenvector is a linear combination of the variables that allows for a common signal to be distilled from the data, thus measuring the collective impact of several indicators at once while conserving degrees of freedom. An eigenvalue tells us the proportion of total variability in a matrix associated with its corresponding eigenvector. Consequently, the eigenvector that corresponds to the highest eigenvalue tells us the dimension (axis) that generates the maximum amount of individual variability in the variables. The next eigenvector is a dimension perpendicular to the first that accounts for the second largest amount of variability, and so on. The eigenvector of the first principal component reveals the combination of variables that has the dominant impact in the overall system. The coefficient loadings of the eigenvector elements are also the correlation coefficients between the principal component and the underlying variables. It is the first principal component that will generally be taken as the synthetic data, proxy variable, latent factor or composite indicator or index. However, such synthetic data, proxy variable, latent factor or composite indicator or index is not limited to only the first principal component, as the second, third, or n^(th) principal component may also be utilized.

For ease of exposition, a brief explanation of eigenvalues and eigenvectors is next presented.

An important mathematical formulation is the characteristic equation of a square matrix. If C is an n by n covariance matrix, the characteristic equation is |C−λl|=0 where λ is a scalar. Solving this equation for λ reveals that the equation is an n^(th) degree polynomial of λ. That is, there are as many λs as there are variables in the covariance matrix. The n λs that are the roots of this polynomial are known as the eigenvalues of C. Because C is symmetric, all the λs will be real numbers (i.e., not complex or imaginary numbers), although some of the λs may be equal to or less than 0. The λs can be solved for in any order, but it is customary to order them from largest to smallest.

To examine what is meant here, let C denote a two by two correlation matrix that has the form:

$\begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}$ the quantity C−λl may thus be written as:

${C \sim {\lambda I}} = {{\begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} - \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix}} = \begin{pmatrix} {1 - \lambda} & \rho \\ \rho & {1 - \lambda} \end{pmatrix}}$ and the determinant is |C−λl|=(1−λ)²−ρ² So the equation that requires solution is (1−λ)²−ρ²=0 which is a quadratic in λ (if we had three variables it would be cubic, and if we had four variables it would be quartic, etc.). Solving the quadratic gives λ=1±ρ The largest root depends on the sign of ρ. For ρ>0, Then λ₁=1+ρ and λ_(s)=1−ρ. For each λ one can define a nonzero vector a such that (C−λl)a=0 The 0 to the right of the equals sign denotes a vector filled with 0s. Any number in a that satisfies this equation is called a latent vector or eigenvector of matrix C. Each eigenvector is associated with its own λ. It is noted that the solution to a is not unique, because if a is multiplied by any scalar, the above equation still holds. Thus, there is an infinite set of values for a, although each solution will be a scalar multiple of any other solution. It is customary to normalize the values of a by imposing the constraint that a′ a=1. A latent vector subject to this constraint is called a normalized latent vector.

Taking the two by two correlation matrix with ρ>0, then

${\left( {C - {\lambda I}} \right)a} = {{\begin{pmatrix} {1 - \lambda} & \rho \\ \rho & {1 - \lambda} \end{pmatrix}\begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix}} = {\begin{pmatrix} {{\left( {1 - \lambda} \right)a_{1}} + {\rho a_{2}}} \\ {{\left( {1 - \lambda} \right)a_{2}} + {\rho a_{1}}} \end{pmatrix} = \begin{pmatrix} {0} \\ 0 \end{pmatrix}}}$ or, by carrying out the multiplication, we find (1−λ)a ₁ +ρa ₂=0 (1−λ)a ₂ +ρa ₁=0 Now taking the largest eigenvalue, λ=1+ρ and substituting gives ρ(a ₂ −a ₁)=0 ρ(a ₁ −a ₂)=0 Thus, all we know is that a₁=a₂. If we let a₁=10, then a₂=10; and if we let a₁=−0.023, then a₂=−0.023. This is what was meant above when it was said that there were an infinite number of solutions where any single solution is a scalar multiple of any other solution. By requiring that a′a=1, we can settle on values of a₁ and a₂. That is, if a₁ ²+a₂ ²=1 and a₁=a₂=a, then 2a²=1 and a=√{square root over (0.5)}.

So the first eigenvector will be a 2 by 1 column vector with both elements equaling √{square root over (0.5)}. For the second eigenvector, we substitute 1−ρ for λ. This gives, ρ(a₁+a₂)=0 Consequently, a₁=−a₂. One of the a's must equal √{square root over (0.5)} and the other must equal −√{square root over (0.5)}. It is immaterial which is positive and which is negative, but it is a frequent convention to make the first one (a₁) positive and the second one (a₂) negative. It is noted that the normalized eigenvectors of a two by two correlation matrix will always take on these values. The actual value of the correlation coefficient is irrelevant as long as it exceeds 0.

The Kalman (Bucy) filter is a recursive algorithm for sequentially updating the one-step ahead estimate of the state mean and variance given new information. Given initial values for the state mean and covariance, values for the system matrices Ξ₁, and observations on y_(t), a Kalman filter may be used to compute one-step ahead estimates of the state and the associated mean-square error matrix, the contemporaneous or filtered state mean and variance, and the one-step ahead prediction, prediction error, and prediction error variance.

In exemplary embodiments of the present disclosure an exemplary Modified Principal Components Analysis-Kalman Filter (MPCA-KF) algorithm differs from conventional Principal Components Analysis (PCA) in several aspects. First, it is calculated recursively, thus eliminating the look-ahead bias inherent in PCA. Second, dynamic on-the-fly programming allows seamless incorporation of new, discontinued, or revised data series, thus resolving the revision issue that plagues economic and financial data. Third, it can be performed on a matrix of pairwise correlations or covariances (in addition to the standard square correlation or covariance matrix), thus allowing the calculation to be performed on imperfect data sets. Fourth, if a balanced panel is desired, missing observations can be filled in by estimating the values by ARIMA, arithmetic, or logarithmic interpolation or extrapolation, as is necessary. Fifth, tattered edges at the end of the sample caused by spontaneous arrival of new data can be smoothed by either: (i) applying a Kalman Filter or (ii) by ARIMA (Autoregressive Integrated Moving Average) or arithmetic or logarithmic extrapolation (estimating series to fill in tattered edges), thus resolving the spontaneous arrival problem associated with economic and financial data, and eliminating the need to discard valuable data observations. Alternatively, the process can include only the first step, whereby a Principal Components Analysis is performed recursively on any of a standard correlation or covariance matrix, or a matrix of pairwise correlations or covariances, in order to eliminate the look-ahead bias that is inherent in PCA. In this case, the recursion requires an initial ramp-up period of an arbitrary length of n observations to begin the process, and at each successive time period the eigenvector loadings are collected such that the end result is a time series of recursively calculated principal components, whereby the eigenvector loadings can serve as index weights. Note that such a calculation can either be calculated in a recursive or rolling fashion. It is understood that such weights range in value between −1 and 1, and collectively sum to a fixed number.

It is noted that there exists the possibility that an eigendecomposition on a matrix of pairwise correlations or covariances (as opposed to the standard PCA method which is an eigendecomposition of a standard square correlation or covariance matrix) may result in a matrix that not positive semidefinite, and the eigendecomposition cannot be performed. In this context it is recalled that a positive semidefinite matrix is a Hermitian matrix all of whose eigenvalues are nonnegative. A square matrix is called Hermitian if it is self-adjoint.

A self-adjoint matrix is a matrix A for which A^(H)≡Ā^(T)=A, where the conjugate transpose is denoted A^(H), the transpose is A^(T), and the complex conjugate is Z. If a matrix is self-adjoint, it is said to be Hermitian. That is, it is noted that the properties of a matrix of pairwise correlations (covariances) are not quite the same as those of a standard correlation (covariance) matrix. In the case that the matrix of pairwise correlations (covariances) is not positive semidefinite, then the data matrix is forced to become balanced or squared by either 1) filling in the missing values by estimation using ARIMA, arithmetic, or logarithmic interpolation or extrapolation; or 2) a Kalman filter is applied.

In exemplary embodiments of the present disclosure such calculation can be performed continually by automatically checking and detecting the shape of the matrix. If spontaneously arrived data have caused a tattered edge at the end of the dataset, then the data can be filled in with estimates, either by ARIMA or arithmetic or logarithmic extrapolation, or the Kalman Filter can be applied. As long as the edge remains tattered, estimates are calculated to square the tattered edge. Once estimated data have completed the tattered edge, the calculation switches back to recursive principal components analysis, and the eigendecomposition performed. Once actual data have arrived, the estimate data can be replaced by the actual values. Once new data arrives to cause a new tattered edge in the next time period, the estimation procedure is resumed until the edge becomes square; the eigendecomposition is performed; then estimates are replaced with actual data, the eigendecomposition performed again and so on, repeating as each new data point arrives. Note that if there is no missing data and all data arrive simultaneously, the calculation can be performed with a recursive or rolling Principal Components Analysis computed using a standard correlation or covariance matrix.

In exemplary embodiments of the present disclosure a calculation can be performed as follows. An initial ramp-up period can be designated, such as, for example, the first 20 observation time periods, whereby initial values are calculated using standard Principal Components Analysis (PCA). In the case that an observation is missing at any time period between time t=1 and t=20, the missing data can be estimated by ARIMA interpolation, or, for example, arithmetic or logarithmic interpolation. Then, at each successive time period after t=20, the principal components can be estimated recursively, adding a set of n observations at each successive time t. That is to say, at each point in time, an eigenvalue decomposition is performed on the standard correlation or covariance matrix, or on the matrix of pairwise correlations or covariances. Data enter the calculation dynamically via on-the-fly programming. That is, at each point in time the program automatically detects which data series are available and performs the calculation on that data set. That is, the calculation is performed at each point in time only on the data that is currently in the Information Set. If a series has been discontinued or introduced in a given time period, the Information Set reflects that change. If a data series has been revised in a given time period, then the revision enters the Information Set at that point in time, and the unrevised data also remains in the information set (or a user can instruct the program to replace the unrevised value with the revised value). In the case that a series has been revised, the newly revised value can be substituted for the previous value, or the unrevised and revised values can enter the calculation concurrently. The key is that the program can automatically detect what is in the Information Set at each point in time, so as to avoid the look-ahead bias that is inherent in standard PCA. In the case newly arrived data causes a tattered edge at time t, either (i) the data can be estimated by ARIMA or arithmetic or logarithmic extrapolation to complete the square matrix, or (ii) a Kalman Filter is applied. Instead of being performed only on a balanced data matrix and discarding newly arrived or revised data, the calculation can performed on the entire data set including the tattered edge, i.e., on an unbalanced data matrix, or data are estimated or a Kalman Filter is applied to fill the tattered edge, if necessary.

Thus, in exemplary embodiments of the present disclosure such an MPCA-KF algorithm can be applied to a standard correlation or covariance matrix, or to a matrix of pairwise correlations (covariances), generated from an unbalanced data matrix with a tattered edge. This allows signal extraction from imperfect datasets, eliminating the need to trim to a square matrix and discard valuable data. As noted herein, at the final tattered edge of newly arrived data, if the matrix of pairwise correlations (covariances) is not positive semidefinite, then either (i) the missing data are filled in by ARIMA, arithmetic, or logarithmic extrapolation; or (ii) a Kalman Filter can be applied.

Alternatively, in exemplary embodiments of the present disclosure, a two-step approach can be employed whereby first the tattered edge of newly arrived data can be truncated to form a square matrix. The modified principal components can be estimated at each point in time recursively and dynamically on either a standard correlation or covariance matrix or a matrix of pairwise correlations or covariances. Then, a Kalman Filter can be employed at the final tattered edge of newly arrived data and the estimation repeated. This two-step procedure maybe performed in a recursive fashion such that a time series of eigenvector loadings is obtained. This technique addresses the general problem of performing an analysis in real time, such as analyzing an economy, where a forecast must be conducted on the basis of data sets that, due to different arrival times, caused for example, by publication lags, result in a tattered edge, i.e., an unbalanced panel, at the end of the sample period.

Alternatively, if there is no missing data and the panel is balanced, i.e., the matrix is square, then a recursive or rolling Principal Components Analysis computed using a standard correlation or covariance matrix such that a time series of eigenvector loadings is obtained.

Thus, the method described above allows for the calculation of a signal in real-time in exemplary embodiments of the present disclosure. The method can correct the look-ahead bias that is inherent in standard PCA by performing the calculation recursively at each point in time. The method can correct the revision problem associated with official economic statistics that are subject to publication revisions because the data enter the calculation dynamically and on the fly. The method can also correct for missing data points by estimating values with ARIMA, arithmetic, or logarithmic interpolation or extrapolation. The method can correct for the spontaneous arrival problem associated with publication lags of official economic data or time zone issues in financial markets, such as the market for foreign exchange where markets close and open at different times around the world.

B. Exemplary Network Environment

FIG. 1 illustrates an example environment in which exemplary embodiments of the present disclosure can be implemented. Relevant asset/securities price data and other data can, for example, be stored on a relational database 20 (as are well known and provided by, for example, IBM, Microsoft Corporation, Oracle and the like) associated with a computer system 10 provided with and running various computational hardware and software applications necessary to generate one or more indicators. Computer system 10 can include, for example, a microprocessor 30, which is understood to include various multiple core processors, various distributed processors, etc., memory (not shown), a storage medium (not shown), input devices (e.g., keyboard, mouse, microphone, etc.) 40, and one or more monitors 50. System 10 can, for example, be operated using a conventional operating system, and can include, for example, a graphical user interface for navigating and controlling various computational aspects of the present disclosure. System 10 can, for example, also be linked to one or more external data source servers 60 that feed system 10 with some or all of the necessary external data for computing the various indicators. Alternatively, as shown in FIG. 1 , a stand-alone workstation 70, including a processor, memory, input devices and storage medium may be used to access database 20, or for example, a combination of database 20 and various external data source servers (not shown) akin to external data source servers 60.

Any suitable hardware and/or software can be used to perform the mechanisms described herein. For example, a general purpose device such as a computer or a special purpose device such as a client, a server, etc. can be used to execute software for performing the mechanisms described herein. Any of these general or special purpose devices can include any suitable components such as a hardware processor (which can be a microprocessor, digital signal processor, a controller, etc.), memory, communication interfaces, display controllers, input devices, etc. This hardware and/or software can be implemented as part of other equipment or can be implemented as stand-alone equipment (which can be coupled to other equipment).

In some embodiments, any suitable computer readable media can be used for storing instructions for performing the processes described herein. For example, in some embodiments, computer readable media can be transitory or non-transitory. For example, non-transitory computer readable media can include media such as magnetic media (such as hard disks, floppy disks, etc.), optical media (such as compact discs, digital video discs, Blu-ray discs, etc.), semiconductor media (such as flash memory, electrically programmable read only memory (EPROM), electrically erasable programmable read only memory (EEPROM), etc.), any suitable media that is not fleeting or devoid of any semblance of permanence during transmission, and/or any suitable tangible media. As another example, transitory computer readable media can include signals on networks, in wires, conductors, optical fibers, circuits, any suitable media that is fleeting and devoid of any semblance of permanence during transmission, and/or any suitable intangible media.

C. Example Operation

FIG. 2 illustrates a basic process according to exemplary embodiments of the present disclosure. In operation, a forecast or trade recommendation generated at 210 can forecast financial quantities in the following general manner. Forecast 210 can generate and/or receive parameters generated by portfolio impact 209 and statistical analysis/econometric modeling 208, which receives automatically, via a network, or manually entered by a user, a set of indicators at 206 which have been processed at 205 and/or receive directly other data which has been processed solely at 206. Data received at 206 can comprise data received from various Internet applications such as, but not limited to, securities price/asset data 201, other transaction data 202, economic and financial data 203, and/or other data 204. In the processing node at 205, the data are cleaned, transformed, normalized and prepared for analysis. If an indicator is to be generated from multiple data sources, a signal is extracted at node 206. A real-time correlation risk index and alerts are generated at 207.

As noted above, where multiple data sources are chosen to construct an indicator, such data sources or keywords can be combined into one composite indicator by assigning weights to each data source after the data are processed and transformed accordingly. In exemplary embodiments of the present disclosure the weights can be assigned either arbitrarily or by some means of optimization, such as, for example, by applying a signal extraction algorithm to find the common signal among the various data. It is understood that such weights range in value between −1 and 1, and collectively sum to a fixed number. Signal extraction algorithm 206 can be, for example, a static, rolling, or recursive Principal Components Analysis which is an eigenvalue decomposition of a covariance or correlation matrix, or a matrix of pairwise correlations and covariances, or a method such as a Kalman Filter, or the Wiener-Kolmogorov (WK) signal extraction formulae, or some other method which modifies or incorporates elements of one or several well-known signal extraction techniques, such as the hybrid MPCA-KF signal extraction algorithm described herein. It is understood that such signal extraction methods may need to be modified due to the unique properties of this type of data. In exemplary embodiments of the present disclosure it is also possible to employ various signal extraction techniques in order to construct various candidate synthetic data, proxy variables or indices or composite indicators and then select the indicator which results in the lowest forecast error of estimated parameters, i.e., select the indicator which results in the best forecast.

In exemplary embodiments of the present disclosure an example system can offer as a choice several well-known signal extraction algorithms, the hybrid MPCA-KF algorithm described above, or some other hybrid or modified signal extraction method. All possible signal extraction methods can be applied concurrently in order to identify the method that results in the lowest forecast error or highest level of predictive power. An example system can run several signal extraction algorithms at once to generate candidate indices or indicators. These candidate variables can then be tested concurrently in a statistical or econometric forecasting model. The example system can then automatically identify the candidate indicator that produces the lowest forecast error or highest level of predictive power, and recommend that indicator as a suitable indicator to the user.

For example, a graph can represent data such as, the correlation risk indicator among various assets trending over time, as shown in FIG. 3 . With reference thereto, FIG. 3 depicts an exemplary graphical display of a correlation risk index. It is understood that such graphical depiction may also include one or more of the following:

(i) calculating and displaying how the correlation risk trends over time;

(ii) calculating and displaying how the correlation risk varies by groups of securities or assets in a portfolio, between asset classes, or among securities comprising a benchmark financial index,

(iii) calculating and displaying how the correlation risk concurrently trends over time and varies by groups of securities or assets in a portfolio, in order to determine if trends are similar or different among different actual or hypothetical groups of securities in a portfolio; (iv) calculating and displaying an overall volume of correlation risk in financial markets per unit of time; (v) calculating and displaying how said correlation risk trends over time and would vary by changing the composition of a portfolio; (vi) calculating and displaying the correlation risk of certain securities or assets, relative to other financial or economic risks; (vii) calculating and displaying the correlation risk of certain securities or assets relative to other subject matter in the same category.

It is also understood that exemplary graphical display may be accompanied by or include the capacity to generate real-time graphical, numerical, auditory, visual or textual alerts regarding current or expected correlation risk of certain securities or assets in a portfolio, wherein said alerts can be in the form of a scrolling ticker displaying levels or changes (numerical or percent) of the correlation risk index, or generating an alert when the correlation risk index surpasses some threshold, wherein said defined threshold level is 5% greater than the average daily level of correlation risk over some time period, or some other quantity, and, wherein the real-time alerts are based on user-configurable conditions or parameters including one or more of:

(i) abnormally positive correlation risk levels;

(ii) abnormally negative correlation risk levels;

(iii) changes in correlation risk levels above or below a pre-selected threshold.

It is also understood that such a system utilizes a graphical user interface comprising a unique combination of (i) data calculation capabilities, (ii) statistical and econometric analysis and (iii) graphical reporting tools to measure and index correlation risk in a portfolio, or financial markets generally, and recommending trades or asset substitutions in a portfolio to change the correlation risk profile of the portfolio, (iv) a scenario analysis module to determine portfolio impact of changes in correlation risk, (v) a securities screening and selection module to identify candidate securities to buy or sell on the basis of correlation risk.

It is also understood that such a system may display in a graphical user interface one or more visualizations of analytic measurements of correlation risk.

Such a system may further comprise displaying one or more of: How correlation risk: (i) trends over time, (ii) varies by groups of securities or assets in a portfolio, securities in a benchmark financial index, or financial markets at large, and (iii) concurrently trends over time and varies by asset classes, groups of securities or assets in a portfolio, securities in a benchmark financial index, or financial markets at large, (iv) a scenario analysis module to determine portfolio impact of changes in correlation risk, (v) a securities screening and selection module to identify candidate securities to buy or sell on the basis of correlation risk.

It is also understood that such a system may comprise displaying correlation risk, in comparison to other benchmark indices or other risks, such as volatility, liquidity, or sentiment. Additionally, such a system may comprise correlation risk, in comparison to the consensus view of current or future economic conditions or asset prices.

Such a system may further comprise displaying how correlation risk concerning any user-configurable collection of securities or assets trends over a user-configurable time period or geographical region, economic sector, industry group, or style classification.

It is understood that such a system may further comprise a graphical user interface that allows a user to configure parameters, such as a wizard which prompts the user to set parameters by which the system calculates, delivers, and displays correlation risk metric for such a user-configurable measure. In addition, such a system may further read portfolio data such as holdings from an online source of account data, such as from an online brokerage account. Alternatively, such portfolio holdings data may be manually entered or uploaded via a file in a format such as Excel or ascii.

To further concretize the above discussion, a an example calculation of a correlation risk index is next described.

Example Calculation—Correlation Risk Indicator and Method of Hedging Correlation Risk

An indicator of correlation risk for a collection of certain assets can be created, as well as for securities comprising benchmark financial index, securities comprising an individual portfolio, or an overall level of correlation risk among asset classes in financial markets that can affect a portfolio.

An indicator of correlation risk among assets or asset classes can be created, where for n assets there exist a total number of unique asset pairs equaling n(n−1)/2. Suppose, for example, we have the following four (4) assets: Equities (measured by the closing value of the S&P 500), Gold (measured by the spot price per ounce), Oil (measured by the spot price per barrel of NYMEX Crude), and Treasury Bonds (measured by the price of a US government 10-year bond). Then, there are 6 possible asset return correlation pairs, denoted by r, as follows:

r_(E,G)=EG=the correlation between returns on equities and gold

r_(E,O)=EO=the correlation between returns on equities and oil

r_(E,B)=EB=the correlation between returns on equities and bonds

r_(G,O)=GO=the correlation between returns on gold and oil

r_(G,B)=GB=the correlation between returns on gold and bonds

r_(O,B)×OB=the correlation between returns on oil and bonds

It is understood that a correlation measure has a de facto time component, inasmuch as the duration of time over which the correlation is measured must be selected such that the measure is meaningful to financial market participants.

A signal extraction algorithm, such as a recursive or rolling Principal Components Analysis described above, can be used to assign weights w_(ij) to each of the correlation values (or, for example, weights can be assigned arbitrarily) to produce a composite indicator of the overall correlation risk in financial markets as follows: CRI (Correlation Risk Index)=w _(E,G) r _(E,G) +w _(E,O) r _(E,O) +w _(E,B) r _(E,B) +w _(G,O) r _(G,O) +w _(G,B) r _(G,B) +w _(O,B) r _(O,B) Alternatively,

${{\sum\limits_{1 \leq i < j \leq n}^{{n({n - 1})}/2}{w_{i,j}r_{i,j}*M}} + S} = {{CRI}\left( {{Correlation}{Risk}{Index}} \right)}$ where n is the number of assets, n(n−1)/2 is the number of unique asset pairs from n assets where 1≤i<j≤n, w_(i,j) is the weight corresponding to the correlation measure for the i-th and j-th assets in a unique asset pair, r_(i,j) is the correlation measure between the i-th and j-th assets in a unique asset pair, M is a multiplier and S is a scalar. It is understood that the multiplier M and scalar S may take on any value, including 1 and 0, respectively. It is understood that such weights range in value between −1 and 1, and collectively sum to a fixed number.

Derivative instruments—futures, options, and options on futures—or other financial instruments can be issued to track the value of each underlying correlation metric, allowing financial market participants to speculate on the value of the correlation indicator for each security. Similarly, an Exchange Traded Fund or other financial instruments can be created to track the value of the index. As the price of each derivative contract on correlation changes, the index would change value in real time as the prices of the underlying assets change. Derivative instruments—futures, options, and options on futures—or other financial instruments can be issued to track the value of the correlation risk index, providing market participants with a method to hedge the risk of correlation between assets in a portfolio and a method to hedge the overall portfolio risk of correlation between asset classes. Similarly, an Exchange Traded Fund or other financial instruments can also be created to track the value of the index.

Example Calculation

Suppose, for example, we have the following hypothetical data representing monthly percent returns on equities, gold, oil, and bonds.

Equities Gold Oil Bonds June-2007 t = 1 8.88 3.44 8.35 −2.73 July-2007 t = 2 9.76 −5.64 −5.07 6.05 August-2007 t = 3 −6.08 −3.46 −4.20 −0.82 September-2007 t = 4 −11.47 0.18 0.92 2.60 October-2007 t = 5 14.52 −0.30 −2.36 −2.68 November-2007 t = 6 −1.01 2.46 5.29 3.95 December-2007 t = 7 1.79 9.60 9.75 −1.35 January-2008 t = 8 −2.95 −8.89 −11.90 2.77 February-2008 t = 9 −2.97 −4.51 2.57 1.47 March-2008 t = 10 7.26 0.92 −4.84 0.15 April-2008 t = 11 −12.52 0.43 −3.29 −18.94 May-2008 t = 12 −8.27 −2.40 2.71 −10.44 June-2008 t = 13 −2.80 −1.57 −2.93 9.89 July-2008 t = 14 3.74 −1.02 1.48 6.32 August-2008 t = 15 −6.71 1.65 1.96 −2.30 September-2008 t = 16 8.15 3.58 0.86 −2.17 October-2008 t = 17 −1.36 −1.28 −3.33 4.39 November-2008 t = 18 2.11 −1.11 2.76 3.44 December-2008 t = 19 3.85 0.31 1.07 −3.44 January-2009 t = 20 18.45 4.89 2.80 3.86 February-2009 t = 21 −10.34 −8.74 −5.31 −0.85 March-2009 t = 22 10.74 1.72 −1.02 3.10 April-2009 t = 23 4.24 −1.27 7.09 3.88 May-2009 t = 24 4.74 2.96 3.03 1.57 June-2009 t = 25 −0.12 −3.35 −7.30 −9.14 July-2009 t = 26 −8.78 4.58 0.64 1.19 August-2009 t = 27 8.36 8.64 8.63 2.01 Sepember-2009 t = 28 −1.36 −12.33 −2.84 5.24 October-2009 t = 29 4.73 5.63 4.84 −3.93 November-2009 t = 30 −0.24 4.47 1.90 1.78

We begin by calculating the 6 asset pairs' returns correlation measures over an initial 6-month time period (m=6) (or some other arbitrary number), from June 2007-November 2007. The estimation window is expanded by one period successively, and the calculation is repeated recursively at m=7, m=8, etc., until m=30. So we now have recursively calculated the asset pairs' returns correlation measures for periods 6 to 30, where the estimation window is expanded by one observation each period, i.e., m=m+1. The following time series of recursively calculated asset pairs' returns correlation measures is obtained:

EG EO EB GO GB OB Jun-07 t = 1 Jul-07 t = 2 Aug-07 t = 3 Sep-07 t = 4 Oct-07 t = 5 Nov-07 t = 6 −0.0249342 −0.0265902 −0.2585826 0.93373232 −0.4111696 −0.2371038 Dec-07 t = 7 −0.0360078 −0.036793 −0.2425717 0.9307845 −0.4548567 −0.3384986 Jan-08 t = 8 0.09947764 0.11179969 −0.2779084 0.95580215 −0.4867384 −0.3934997 Feb-08 t = 9 0.14095282 0.08734425 −0.2826544 0.88921399 −0.4839709 −0.3838573 Mar-08 t = 10 0.1605178 0.02480602 −0.295755 0.83311182 −0.4889561 −0.3477549 Apr-08 t = 11 0.10923528 0.09226848 0.33348525 0.81346403 −0.2621948 −0.0126216 May-08 t = 12 0.13619456 0.04592749 0.40778086 0.78351706 −0.1934196 −0.0696561 Jun-08 t = 13 0.13997027 0.05595248 0.32893585 0.78285614 −0.196991 −0.1174974 Jul-08 t = 14 0.13644551 0.06688026 0.35016912 0.77852813 −0.1943611 −0.0908244 Aug-08 t = 15 0.10174229 0.04387599 0.35705066 0.7810199 −0.202549 −0.0976385 Sep-08 t = 16 0.16062035 0.05409369 0.32566538 0.76752443 −0.2118156 −0.1001958 Oct-08 t = 17 0.16245583 0.05949289 0.313058 0.76592717 −0.2177902 −0.1239383 Nov-08 t = 18 0.15908981 0.06809349 0.31865694 0.75268847 −0.2211531 −0.1039426 Dec-08 t = 19 0.1629873 0.07358772 0.29966247 0.75322188 −0.2246269 −0.1092966 Jan-09 t = 20 0.28005536 0.12507861 0.32904441 0.75105288 −0.1699447 −0.089229 Feb-09 t = 21 0.36854471 0.18572689 0.32316895 0.76025763 −0.1401703 −0.0795672 Mar-09 t = 22 0.38295765 0.16973999 0.33867624 0.75036113 −0.1258781 −0.0833155 Apr-09 t = 23 0.37812761 0.18525811 0.34512759 0.70229488 −0.1301467 −0.0389456 May-09 t = 24 0.38614264 0.19371792 0.34744741 0.70740422 −0.1203571 −0.0331856 Jun-09 t = 25 0.38694859 0.19543689 0.3408623 0.71177449 −0.0667263 0.06365804 Jul-09 t = 26 0.30681888 0.18314141 0.32051198 0.69722821 −0.0548147 0.06468804 Aug-09 t = 27 0.34702891 0.22770434 0.32657836 0.73353953 −0.0243163 0.08267382 Sep-09 t = 28 0.33274532 0.23268104 0.3112572 0.69601082 −0.1018275 0.0611322 Oct-09 t = 29 0.3424034 0.24308945 0.29557983 0.70654222 −0.1288171 0.03668656 Nov-09 t = 30 0.33134619 0.24067005 0.29314989 0.70461442 −0.1173969 0.03963129

Next, we perform a recursive Principal Components Analysis, that is, we perform a recursive eigenvalue decomposition on the asset pairs' returns correlation measures we have just generated. We set an initial ramp-up period of 20 (or some other arbitrary number) observations to begin the calculation at June 2009, again expanding the estimation window by one observation at each time period, such that m=m+1. At each period, we perform a principal components analysis, that is, we obtain a correlation matrix, perform the eigenvalue decomposition, and obtain the eigenvector loadings, also known as the weights. We do the analysis recursively for the rest of the sample period after the ramp-up window, from June 2009 to November 2009. We obtain the following output:

JUNE 2009 obs EG EO EB GO GB OB 2007M11 −0.024934 −0.02659 −0.258583 0.933732 −0.41117 −0.237104 2007M12 −0.036008 −0.036793 −0.242572 0.930785 −0.454857 −0.338499 2008M01 0.099478 0.1118 −0.277908 0.955802 −0.486738 −0.3935 2008M02 0.140953 0.087344 −0.282654 0.889214 −0.483971 −0.383857 2008M03 0.160518 0.024806 −0.295755 0.833112 −0.488956 −0.347755 2008M04 0.109235 0.092268 0.333485 0.813464 −0.262195 −0.012622 2008M05 0.136195 0.045927 0.407781 0.783517 −0.19342 −0.069656 2008M06 0.13997 0.055952 0.328936 0.782856 −0.196991 −0.117497 2008M07 0.136446 0.06688 0.350169 0.778528 −0.194361 −0.090824 2008M08 0.101742 0.043876 0.357051 0.78102 −0.202549 −0.097639 2008M09 0.16062 0.054094 0.325665 0.767524 −0.211816 −0.100196 2008 M10 0.162456 0.059493 0.313058 0.765927 −0.21779 −0.123938 2008M11 0.15909 0.068093 0.318657 0.752688 −0.221153 −0.103943 2008M12 0.162987 0.073588 0.299662 0.753222 −0.224627 −0.109297 2009M01 0.280055 0.125079 0.329044 0.751053 −0.169945 −0.089229 2009M02 0.368545 0.185727 0.323169 0.760258 −0.14017 −0.079567 2009M03 0.382958 0.16974 0.338676 0.750361 −0.125878 −0.083316 2009M04 0.378128 0.185258 0.345128 0.702295 −0.130147 −0.038946 2009M05 0.386143 0.193718 0.347447 0.707404 −0.120357 −0.033186 2009M06 0.386949 0.195437 0.340862 0.711774 −0.066726 0.063658

Cumulative Cumulative Number Value Difference Proportion Value Proportion 1 4.775417 3.825476 0.7959 4.775417 0.7959 2 0.949941 0.793303 0.1583 5.725358 0.9542 3 0.156638 0.087276 0.0261 5.881996 0.9803 4 0.069362 0.029596 0.0116 5.951359 0.9919 5 0.039766 0.030891 0.0066 5.991125 0.9985 6 0.008875 — 0.0015 6 1

Eigenvectors (loadings): Variable PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 EG 0.382757 0.542334 −0.26554 −0.186681 0.336375 0.583834 EO 0.350614 0.626631 0.436045 0.25554 −0.341537 −0.335141 EB 0.41211 −0.40025 0.088635 0.698276 −0.078336 0.410345 GO −0.434112 0.09072 0.738591 0.056925 0.373373 0.33934 GB 0.443407 −0.192704 0.139397 −0.030324 0.72199 −0.473956 OB 0.41913 −0.328088 0.408104 −0.638827 −0.321544 0.196595

Ordinary correlations: EG EO EB GO GB OB EG 1 EO 0.93594 1 EB 0.53541 0.470027 1 GO −0.771447 −0.627468 −0.875739 1 GB 0.712987 0.628286 0.942383 −0.910519 1 OB 0.585084 0.526796 0.926032 −0.856649 0.947743 1

JULY 2009 obs EG EO EB GO GB OB 2007M11 −0.024934 −0.02659 −0.258583 0.933732 −0.41117 −0.237104 2007M12 −0.036008 −0.036793 −0.242572 0.930785 −0.454857 −0.338499 2008M01 0.099478 0.1118 −0.277908 0.955802 −0.486738 −0.3935 2008M02 0.140953 0.087344 −0.282654 0.889214 −0.483971 −0.383857 2008M03 0.160518 0.024806 −0.295755 0.833112 −0.488956 −0.347755 2008M04 0.109235 0.092268 0.333485 0.813464 −0.262195 −0.012622 2008M05 0.136195 0.045927 0.407781 0.783517 −0.19342 −0.069656 2008M06 0.13997 0.055952 0.328936 0.782856 −0.196991 −0.117497 2008M07 0.136446 0.06688 0.350169 0.778528 −0.194361 −0.090824 2008M08 0.101742 0.043876 0.357051 0.78102 −0.202549 −0.097639 2008M09 0.16062 0.054094 0.325665 0.767524 −0.211816 −0.100196 2008M10 0.162456 0.059493 0.313058 0.765927 −0.21779 −0.123938 2008M11 0.15909 0.068093 0.318657 0.752688 −0.221153 −0.103943 2008M12 0.162987 0.073588 0.299662 0.753222 −0.224627 −0.109297 2009M01 0.280055 0.125079 0.329044 0.751053 −0.169945 −0.089229 2009M02 0.368545 0.185727 0.323169 0.760258 −0.14017 −0.079567 2009M03 0.382958 0.16974 0.338676 0.750361 −0.125878 −0.083316 2009M04 0.378128 0.185258 0.345128 0.702295 −0.130147 −0.038946 2009M05 0.386143 0.193718 0.347447 0.707404 −0.120357 −0.033186 2009M06 0.386949 0.195437 0.340862 0.711774 −0.066726 0.063658 2009M07 0.306819 0.183141 0.320512 0.697228 −0.054815 0.064688

Cumulative Cumulative Number Value Difference Proportion Value Proportion 1 4.8206 3.927779 0.8034 4.8206 0.8034 2 0.892821 0.744519 0.1488 5.713421 0.9522 3 0.148302 0.064629 0.0247 5.861724 0.977 4 0.083674 0.046238 0.0139 5.945397 0.9909 5 0.037436 0.020269 0.0062 5.982833 0.9971 6 0.017167 — 0.0029 6 1

Eigenvectors (loadings): Variable PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 EG 0.384662 0.541.829 −0.294595 −0.035598 0.418863 0.5431 EO 0.358946 0.616657 0.434272 0.210449 −0.402849 −0.309393 EB 0.404565 −0.424869 −0.007167 0.759435 −0.102049 0.261929 GO −0.433785 0.093278 0.697597 0.246333 0.415677 0.288135 GB 0.442024 −0.191183 0.156405 −0.062187 0.660451 −0.550944 OB 0.419482 −0.316845 0.462025 −0.559603 −0.207052 0.392622

Ordinary correlations: EG EO EB GO GB OB EG 1 EO 0.935105 1 EB 0.543544 0.479175 1 GO −0.781251 −0.657773 −0.866752 1 GB 0.725732 0.661536 0.925459 −0.91779 1 OB 0.606465 0.572341 0.904788 −0.868583 0.95272 1

AUGUST 2009 obs EG EO EB GO GB OB 2007M11 −0.024934 −0.02659 −0.258583 0.933732 −0.41117 −0.237104 2007M12 −0.036008 −0.036793 −0.242572 0.930785 −0.454857 −0.338499 2008M01 0.099478 0.1118 −0.277908 0.955802 −0.486738 −0.3935 2008M02 0.140953 0.087344 −0.282654 0.889214 −0.483971 −0.383857 2008M03 0.160518 0.024806 −0.295755 0.833112 −0.488956 −0.347755 2008M04 0.109235 0.092268 0.333485 0.813464 −0.262195 −0.012622 2008M05 0.136195 0.045927 0.407781 0.783517 −0.19342 −0.069656 2008M06 0.13997 0.055952 0.328936 0.782856 −0.196991 −0.117497 2008M07 0.136446 0.06688 0.350169 0.778528 −0.194361 −0.090824 2008M08 0.101742 0.043876 0.357051 0.78102 −0.202549 −0.097639 2008M09 0.16062 0.054094 0.325665 0.767524 −0.211816 −0.100196 2008M10 0.162456 0.059493 0.313058 0.765927 −0.21779 −0.123938 2008M11 0.15909 0.068093 0.318657 0.752688 −0.221153 −0.103943 2008M12 0.162987 0.073588 0.299662 0.753222 −0.224627 −0.109297 2009M01 0.280055 0.125079 0.329044 0.751053 −0.169945 −0.089229 2009M02 0.368545 0.185727 0.323169 0.760258 −0.14017 −0.079567 2009M03 0.382958 0.16974 0.338676 0.750361 −0.125878 −0.083316 2009M04 0.378128 0.185258 0.345128 0.702295 −0.130147 −0.038946 2009M05 0.386143 0.193718 0.347447 0.707404 −0.120357 −0.033186 2009M06 0.386949 0.195437 0.340862 0.711774 −0.066726 0.063658 2009M07 0.306819 0.183141 0.320512 0.697228 −0.054815 0.064688 2009M08 0.347029 0.227704 0.326578 0.73354 −0.024316 0.082674

Cumulative Cumulative Number Value Difference Proportion Value Proportion 1 4.854529 4.02067 0.8091 4.854529 0.8091 2 0.833859 0.658473 0.139 5.688388 0.9481 3 0.175386 0.090993 0.0292 5.863773 0.9773 4 0.084393 0.0503 0.0141 5.948166 0.9914 5 0.034092 0.016351 0.0057 5.982259 0.997 6 0.017741 — 0.003 6 1

Eigenvectors (loadings): Variable PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 EG 0.387955 0.536237 −0.325528 −0.03 0.416643 0.530547 EO 0.365955 0.61081 0.371369 0.282301 −0.425737 −0.306801 EB 0.398487 −0.461489 −0.13683 0.748331 −0.089525 0.203716 GO −0.429973 0.116523 0.641119 0.381344 0.389259 0.305885 GB 0.441354 −0.173294 0.226094 −0.076895 0.66002 −0.531524 OB 0.42084 −0.287715 0.524635 −0.456148 −0.223589 0.454762

Ordinary correlations: EG EO EB GO GB OB EG 1 EO 0.931486 1 EB 0.550695 0.481984 1 GO −0.786842 −0.660987 −0.867993 1 GB 0.74539 0.702026 0.906253 −0.90926 1 OB 0.636239 0.62517 0.885746 −0.862569 0.957697 1

SEPTEMBER 2009 obs EG EO EB GO GB OB 2007M11 −0.024934 −0.02659 −0.258583 0.933732 −0.41117 −0.237104 2007M12 −0.036008 −0.036793 −0.242572 0.930785 −0.454857 −0.338499 2008M01 0.099478 0.1118 −0.277908 0.955802 −0.486738 −0.3935 2008M02 0.140953 0.087344 −0.282654 0.889214 −0.483971 −0.383857 2008M03 0.160518 0.024806 −0.295755 0.833112 −0.488956 −0.347755 2008M04 0.109235 0.092268 0.333485 0.813464 −0.262195 −0.012622 2008M05 0.136195 0.045927 0.407781 0.783517 −0.19342 −0.069656 2008M06 0.13997 0.055952 0.328936 0.782856 −0.196991 −0.117497 2008M07 0.136446 0.06688 0.350169 0.778528 −0.194361 −0.090824 2008M08 0.101742 0.043876 0.357051 0.78102 −0.202549 −0.097639 2008M09 0.16062 0.054094 0.325665 0.767524 −0.211816 −0.100196 2008M10 0.162456 0.059493 0.313058 0.765927 −0.21779 −0.123938 2008M11 0.15909 0.068093 0.318657 0.752688 −0.221153 −0.103943 2008M12 0.162987 0.073588 0.299662 0.753222 −0.224627 −0.109297 2009M01 0.280055 0.125079 0.329044 0.751053 −0.169945 −0.089229 2009M02 0.368545 0.185727 0.323169 0.760258 −0.14017 −0.079567 2009M03 0.382958 0.16974 0.338676 0.750361 −0.125878 −0.083316 2009M04 0.378128 0.185258 0.345128 0.702295 −0.130147 −0.038946 2009M05 0.386143 0.193718 0.347447 0.707404 −0.120357 −0.033186 2009M06 0.386949 0.195437 0.340862 0.711774 −0.066726 0.063658 2009M07 0.306819 0.183141 0.320512 0.697228 −0.054815 0.064688 2009M08 0.347029 0.227704 0.326578 0.73354 −0.024316 0.082674 2009M09 0.332745 0.232681 0.311257 0.696011 −0.101828 0.061132

Cumulative Cumulative Number Value Difference Proportion Value Proportion 1 4.885205 4.086723 0.8142 4.885205 0.8142 2 0.798482 0.628432 0.1331 5.683687 0.9473 3 0.17005 0.083986 0.0283 5.853737 0.9756 4 0.086064 0.043356 0.0143 5.939801 0.99 5 0.042708 0.025217 0.0071 5.982509 0.9971 6 0.017491 — 0.0029 6 1

Eigenvectors (loadings): Variable PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 EG 0.389349 0.529186 −0.365196 0.054464 0.46021 0.469299 EO 0.369693 0.605215 0.371935 0.249756 −0.478581 −0.259401 EB 0.393914 −0.488903 −0.192533 0.702762 −0.19877 0.188022 GO −0.430319 0.106521 0.575308 0.5138 0.381657 0.250693 GB 0.4.39838 −0.179132 0.204906 0.018028 0.608142 −0.601919 OB 0.421822 −0.266865 0.564146 −0.420078 −0.064495 0.50196

Ordinary correlations: EG EO EB GO GB OB EG 1 EO 0.925444 1 EB 0.555548 0.481294 1 GO −0.79724 −0.6872 −0.859844 1 GB 0.755274 0.711442 0.903571 −0.911745 1 OB 0.655414 0.658551 0.874232 −0.871686 0.956584 1

OCTOBER 2009 obs EG EO EB GO GB OB 2007M11 −0.024934 −0.02659 −0.258583 0.933732 −0.41117 −0.237104 2007M12 −0.038008 −0.036793 −0.242572 0.930785 −0.454857 −0.338499 2008M01 0.099478 0.1118 −0.277908 0.955802 −0.486738 −0.3935 2008M02 0.140953 0.087344 −0.282654 0.889214 −0.483971 −0.383857 2008M03 0.160518 0.024806 −0.295755 0.833112 −0.488956 −0.347755 2008M04 0.109235 0.092288 0.333485 0.813484 −0.262195 −0.012622 2008M05 0.136195 0.045927 0.407781 0.783517 −0.19342 −0.069656 2008M06 0.13997 0.05S952 0.328936 0.782856 −0.196991 −0.117497 2008M07 0.136446 0.06688 0.350169 0.778528 −0.194361 −0.090824 2008M08 0.101742 0.043876 0.357051 0.78102 −0.202549 −0.097639 2008M09 0.16062 0.054094 0.325665 0.767524 −0.211816 −0.100196 2008M10 0.162458 0.059493 0.313058 0.765927 −0.21779 −0.123938 2008M11 0.15909 0.068093 0.318657 0.752688 −0.221153 −0.103943 2008M12 0.162987 0.073588 0.299682 0.753222 −0.224627 −0.109297 2009M01 0.280055 0.125079 0.329044 0.751053 −0.169945 −0.089229 2009M02 0.36854S 0.18S727 0.323169 0.760258 −0.14017 −0.079567 2009M03 0.382958 0.16974 0.338676 0.750361 −0.125878 −0.083316 2009M04 0.378128 0.185258 0.345128 0.702295 −0.130147 −0.038946 2009M05 0.386143 0.193718 0.347447 0.707404 −0.120357 −0.033186 2009M06 0.386949 0.19S437 0.340862 0.711774 −0.066726 0.063858 2009M07 0.306819 0.183141 0.320512 0.697228 −0.054815 0.064688 2009M08 0.347029 0.227704 0.326578 0.73354 −0.024316 0.082674 2009M09 0.332745 0.232681 0.311257 0.696011 −0.101828 0.061132 2009M10 0.342403 0.243089 0.29558 0.706542 −0.128817 0.036687

Cumulative Cumulative Number Value Difference Proportion Value Proportion 1 4.900675 4.118612 0.8168 4.900675 0.8168 2 0.782064 0.615564 0.1303 5.682739 0.9471 3 0.1665 0.081348 0.0277 5.849239 0.9749 4 0.085152 0.036373 0.0142 5.934391 0.9891 5 0.048779 0.031948 0.0081 5.98317 0.9972 6 0.01683 — 0.0028 6 1

Eigenvectors (loadings): Variable PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 EG 0.390805 0.520303 −0.393701 0.101498 0.440717 0.465856 EO 0.370914 0.604355 0.370442 0.234242 −0.483881 −0.266347 EB 0.39105 −0.507936 −0.216502 0.655203 −0.28297 0.181232 GO −0.430873 0.096859 0.528924 0.588029 0.344984 0.245794 GB 0.438459 −0.187833 0.181082 0.082187 0.610357 −0.600327 OB 0.422939 −0.247745 0.59021 −0.391159 −0.002537 0.508316

Ordinary correlations: EG EO E8 GO GB OB EG 1 EO 0.92154S 1 EB 0.557449 0.47633 1 GO −0.806043 −0.702322 −0.854475 1 GB 0.760562 0.709313 0.902682 −0.912214 1 OB 0.671067 0.678076 0.867424 −0.87738 0.955027 1

NOVEMBER 2009 obs EG EO EB GO GB OB 2007M11 −0.024934 −0.02659 −0.258583 0.933732 −0.41117 −0.237104 2007M12 −0.036008 −0.036793 −0.242572 0.930785 −0.454857 −0.338499 2008M01 0.099478 0.1118 −0.277908 0.955802 −0.486738 −0.3935 2008M02 0.140953 0.087344 −0.282654 0.889214 −0.483971 −0.383857 2008M03 0.160518 0.024806 −0.295755 0.833112 −0.488956 −0.347755 2008M04 0.109235 0.092268 0.333485 0.813464 −0.262195 -0.012622 2008M05 0.136195 0.045927 0.407781 0.783517 −0.19342 −0-069656 2008M06 0.13997 0.055952 0.328936 0.782856 −0.196991 −0.117497 2008M07 0.136446 0.06688 0.350169 0.778528 −0.194361 −0.090824 2008M08 0.101742 0.043876 0.357051 0.78102 −0.202549 −0.037639 2008M09 0.16062 0.054094 0.325665 0.767524 −0.211816 −0.100196 2008M10 0.162456 0.059493 0.313058 0.765927 −0.21779 −0.123938 2008M11 0.15909 0.068093 0.318657 0.752688 −0.221153 −0.103943 2008M12 0.162987 0.073588 0.299662 0.753222 −0.224627 −0.109297 2009M01 0.280055 0.125079 0.329044 0.751.053 −0.169945 −0-089229 2009M02 0.368545 0.185727 0.323169 0.760258 −0.14017 −0.079567 2009M03 0.382958 0.16974 0.338676 0.750361 −0.125878 −0.083316 2009M04 0.378128 0.185258 0.345128 0.702295 −0.130147 −0.038946 2009M05 0.386143 0.193718 0.347447 0.707404 −0.120357 −0-033186 2009M06 0.386949 0.195437 0.340862 0.711774 −0.066726 0.063658 2009M07 0.306819 0.183141 0.320512 0.697228 −0.054815 0.064688 2009M08 0.347029 0.227704 0.326578 0.73354 −0.024316 0.082674 2009M09 0.332745 0.232681 0.311257 0.696011 −0.101828 0.061132 2009M10 0.342403 0.243089 0.29558 0.706542 −0.128817 0.036687 2009M11 0.331346 0.24067 0.29315 0.704614 −0.117397 0.039631

Cumulative Cumulative Number Value Difference Proportion Value Proportion 1 4.91657 4.150551 0.8194 4.91657 0.8194 2 0.756018 0.601334 0.1277 5.682588 0.9471 3 0.164684 0.07976 0.0274 5.847272 0.9745 4 0.084925 0.033687 0.0142 5.932197 0.9887 5 0.051238 0.034672 0.0085 5.983435 0.9972 6 0.016555 — 0.0028 6 1

Eigenvectors (loadings): Variable PC 1 PC 2 PC 3 PC 4 PC 5 PC 6 EG 0.391828 0.512324 −0.425156 0.139569 0.426975 0.448832 EO 0.372455 0.602103 0.368137 0.219161 −0.499117 −0.257049 EB 0.388318 −0.525441 −0.24088 0.603161 −0.347206 0.175335 GO −0.431179 0.089161 0.478019 0.655574 0.306985 0.231553 GB 0.437683 −0.191434 0.162599 0.135369 0.591384 −0.614238 OB 0.423646 −0.233047 0.608896 −0.347253 0.063617 0.520411

Ordinary correlations: EG EC EB GO GB OB EG 1 EO 0.917803 1 EB 0.559589 0.473498 1 GO −0.812915 −0.716106 −0.849261 1 GB 0.766637 0.71306 0.900851 −0.913644 1 OB 0.683187 0.694904 0.86106 −0.882415 0.954762 1

We see that the first principal component summarizes approximately 80% of the variation in the system. We then create a time series of the eigenvector loadings pertaining to the first principal component, over the six month time period during which the principal components were calculated.

Eigenvector Loading Summary

Eigenvector Loadings Summay PC 1 W_EG W_EO W_EB W_GO W_GB W_OB 2009M06 0.382757 0.350614 0.41211 −0.434112 0.443407 0.41913 2009M07 0.384662 0.358946 0.404565 −0.433785 0.442024 0.419482 2009M08 0.387955 0.365955 0.398487 −0.429973 0.441354 0.42084 2009M09 0.389349 0.369693 0.393914 0.430319 0.439838 0.421822 2009M10 0.390805 0.370914 0.39105 −0.430873 0.438459 0.422939 2009M11 0.391828 0.372455 0.388318 −0.431179 0.437683 0.423646

Substitution of Values and Index Calculation PC 1 (W_EG * EG) + (W_EO * EO) + (W_EB * EB) + (W_GO * 2009M06 = (0.38276 * 0.38694259) + (0.35061 * 0.19543689) + 0.41211 *  0.3408623) + (−0.4341 * 2009M07 = (0.38466 * 0.30621288) + (0.35895 * 0.18314141) + 0.40457 * 0.32051198) +  (0.4338 * 2009M08 = (0.38796 * 0.34702891) + (0.36596 * 0.22770434) + 0.39849 * 0.32657836) +  (−0.43 * 2009M09 = (0.32935 * 0.33274532) + (0.36969 * 0.23268104) + 0.39391 *  0.3112572) + (−0.4303 * 2009M10 = (0.39081 *  0.3424034) + (0.37091 * 0.24308945) + 0.39105 * 0.29557983) + (−0.4309 * 2009M11 = (0.39183 * 0.33134619) + (0.37246 * 0.24067005) + 0.38832 * 0.29314989) +  (0.4312 * PC 1 GO) + (W_GB * GB) + (W_OB * OB) = WeightedSum 2009M06 = 0.71177449) + (0.44341 *  0.0667263) + (0.41913 * 0.06365804) =  0.045207215 2009M07 = 0.69722821) + (0.44202 * −0.0548147) + (0.41948 * 0.06458804) =  0.013886271 2009M08 = 0.73353953) + (0.44135 * −0.0243163) + (0.42084 * 0.08267382) =  0.056756544 2009M09 = 0.69601082) + (043984 * −0.1018275) + (0.42182 *  0.0611322) =  0.01967579 2009M10 = 0.70654222) + (0.43846 * −0.1288171) + (0.42294 * 0.03668656) = −0.005830063 2009M11 = 0.70461442) + (0.43768 * −0.1173959) + (0.42365 * 0.03953129) = −0.005103072

Correlation Risk Index PC 1 WeightedSum *100 +100 Index Value 2009M06 0.045207215 4.5207215 104.520721 104.520721 2009M07 0.013886271 1.38862706 101.388627 101.388627 2009M08 0.056756544 5.67565436 105.675654 105.675654 2009M09 0.01967579 1.96757903 101.967579 101.967579 2009M10 −0.005830063 −0.5830063 99.4169937 99.4169937 2009M11 −0.005103072 −0.5103072 99.4896928 99.4896928

Once the time series of eigenvector loadings from the first and/or n-th principal component have been obtained, and the Correlation Risk Index has been calculated, it can be analyzed on a standard analytical platform which can include graphs and summary statistics, as well as basic econometric analysis, such as OLS, MLE, GMM, etc., to determine its impact on a portfolio or predictive power for future returns. The Program allows a user to import other data to use in the analysis, for comparisons, making forecasts, etc. If the calculation utilizes real-time data via an API feed, the program can perform the calculation dynamically (in near real-time), and generate a numerical value for the index dynamically (in near real-time), as well as a graph, which updates dynamically (in near real-time), or displays the level of correlation risk in a scrolling ticker. Once the level of correlation risk affecting broad asset classes, securities comprising an index or individual portfolio or financial markets in general has been quantified and indexed, financial instruments such as options and futures can be issued to track the level of market-wide, index-specific, or portfolio-specific correlation risk such that investors can hedge against this risk. 

What is claimed:
 1. A computer-implemented method, the method comprising: accessing data comprising a plurality of returns corresponding to each of a plurality of assets for a time period, wherein the data comprises a time series data for each asset in the plurality of returns; performing an analysis of the time series data to find a correlation measure between at least two assets from the plurality of assets over a defined time window in the time period; rolling the defined time window by a set number of time periods to identify a rolled time window; updating the analysis, on an on-going basis, for the rolled time window to find an updated correlation measure between the at least two assets from the plurality of assets over the rolled time window in the time period; and displaying, via a graphical user interface, the updated correlation measure over the rolled time window.
 2. The method of claim 1, wherein the correlation measure is a measure of a co-movement between at least two assets from the plurality of assets, the measure of a co-movement comprising at least one of: a correlation metric, a cointegration metric, a covariance metric, a causality metric, and a dependency metric.
 3. The method of claim 2, wherein when the measure of a co-movement is a correlation metric, the correlation metric comprises at least one of: a simple correlation, a multivariate correlation, a Spearman correlation, a Pearson correlation, a Kendall correlation, a time-varying correlation, a dynamic correlation, and a stochastic correlation.
 4. The method of claim 1, wherein the analysis comprises: classifying each asset from the plurality of assets into an asset class to obtain a plurality of asset classes; forming pairs of assets within a respective asset class from the plurality of asset classes; determining the correlation measure for each pair of assets within a respective asset class for each rolled time window in the time period to obtain determined correlation measures; computing, based on the determined correlation measures for the pairs of assets within a respective asset class, a mean correlation value for each asset class for each rolled time window in the time period; and displaying, via a graphical user interface, at least one mean correlation value for at least one asset class for each rolled time window in the time period.
 5. The method of claim 1, wherein the analysis comprises: determining the correlation measure for the at least two assets from the plurality of assets for each rolled time window in the time period; obtaining a plurality of time series of correlation measures for the at least two assets from the plurality of assets by aggregating each rolled time window in the time period, wherein the plurality of time series of correlation measures are a measure of a co-movement across the plurality of assets; performing a principal components analysis on the plurality of time series of correlation measures for the at least two assets from the plurality of assets; and determining, based on the principal components analysis, at least one statistical risk factor in the time series data that explains part of the co-movement across the plurality of assets.
 6. The method of claim 1, further comprising: determining, based on a statistical risk factor, a variability of the updated correlation measure over the rolled time window; and displaying, via a graphical user interface, the variability of the updated correlation measure over the rolled time window.
 7. A computer-implemented method, the method comprising: creating a plurality of hypothetical portfolios of securities, wherein each hypothetical portfolio from the plurality of hypothetical portfolios is formed to according to a set of rules that conform to a respective style of investment represented by a respective style factor; conducting a plurality of hypothetical securities trades in each of the hypothetical portfolios according to the respective set of rules; calculating a hypothetical return for each hypothetical portfolio from the plurality of hypothetical portfolios over a plurality of points in time in a time period; generating a style index to track the hypothetical return for each hypothetical portfolio from the plurality of hypothetical portfolios over the plurality of points in time in the time period; displaying, on an on-going basis via a graphical user interface, a plurality of style indices from the style index generated for each hypothetical portfolio for the plurality of points in time in the time period; quantifying a correlation measure between at least two style indices from the plurality of style indices over the plurality of points in time in the time period; generating at least one correlation index of the correlation measure between the at least two style indices from the plurality of style indices over the plurality of points in time in the time period; and displaying, on an on-going basis via a graphical user interface, the at least one correlation index for the plurality of points in time in the time period.
 8. The method of claim 7, further comprising: tracking at least one of a style index and a correlation index from the at least one correlation index between the at least two style indices with a financial instrument issued on at least one of a value of the style index and value of the correlation index between the at least two style indices, respectively.
 9. The method of claim 8, wherein the financial instrument comprises at least one of: a derivative instrument, a future, an option, an option on a future, and an Exchange Traded Fund.
 10. A computer-implemented method, the method comprising: accessing data for a time period, the data comprising information on at least one of: (i) a plurality of assets, (ii) a plurality of asset classes, (iii) economic data, and (iv) financial data; identifying a correlation in the data for a plurality of points in time in the time period, wherein the correlation comprises a relationship between the information on at least one of: (i) assets from the plurality of assets, (ii) asset classes from the plurality of asset classes, (iii) the economic data, (iv) the financial data, (v) an asset from the plurality of assets or an asset class from the plurality of asset classes and the economic data, (vi) an asset from the plurality of assets or an asset class from the plurality of asset classes and the financial data, and (vii) the economic data and the financial data; and generating, on an on-going basis, an indicator for the correlation over the plurality of points in time in the time period.
 11. The method of claim 10, wherein the identifying comprises calculating a metric for the relationship and wherein the metric is an indicator and the indicator comprises at least one of: a correlation indicator, a cointegration indicator, a covariance indicator, a causality indicator, and a dependency indicator.
 12. The method of claim 11, wherein when the metric is a correlation indicator, the correlation indicator comprises at least one of: a simple correlation, a multivariate correlation, a Spearman correlation, a Pearson correlation, a Kendall correlation, a time-varying correlation, a dynamic correlation, and a stochastic correlation.
 13. The method of claim 10, further comprising: expanding the time period; identifying a time series of correlations in the data for a plurality of points in time in the expanded time period; and generating, on an on-going basis, an updated indicator for the time series of correlations over the plurality of points in time in the expanded time period.
 14. The method of claim 10, wherein the information comprises at least one of: return data on an asset from the plurality of assets, return data on an asset class from the plurality of asset classes, economic statistics published by a government agency, economic statistics related to a sector, economic statistics pertaining to an industry, financial information pertaining to a sector, financial information pertaining to at least one company within a sector, financial information pertaining to an industry, financial information pertaining to at least one company within an industry, financial information pertaining to an identified market index, financial information pertaining to at least one company within an identified market index, financial information pertaining to a portfolio of assets, and financial information pertaining to at least one asset from a portfolio of assets.
 15. The method of claim 14, wherein when the information comprises return data, the return data is based on at least one of: a percent change of return on at least one asset in the plurality of assets for the time period, and a log return on at least one asset in the plurality of assets for the time period.
 16. The method of claim 11, further comprising: selecting at least one candidate asset from the plurality of assets based on the indicator.
 17. The method of claim 10, wherein the data further comprises at least one of: moments of a statistical distribution of the data, a statistical measure of the data, a transformation of the data, and a property of the data over the time period.
 18. The method of claim 10, further comprising: displaying the indicator on a graphical user interface on an on-going basis.
 19. The method of claim 10, wherein the identifying a correlation in the data comprises the data entering a calculation dynamically and on-the-fly.
 20. The method of claim 19, wherein the data entering comprises, at each point in time in the time period, detecting at least two time series of data from the data that is available and performing the calculation on the at least two time series of data. 